One hasn't budded at all this spring.
At this point, that one is probably dead. Depending on the species, green will show through the young bark when it is alive. If there is no green showing, scratch lightly through the bark on the trunk near the ground. If the cambium layer is wet and green or white, the tree is alive. If it is very dry, or brown to black, the tree is dead. If the terminal growth was alive, the tree would have begun to bud by now. This tree, even if not quite dead, won't survive well and should be removed.
The other tree has a few small leaves on one of the lowest branches.
That signifies that the rest of the branches are dead. If you want to try and save it, don't transplant it. that will put it in shock, and won't help the tree. You can cut back all of the dead from the tree, to help with appearances, but it should be for the most part left alone. This tree doesn't have an established enough root system to safely fertilize with most products yet, but water as necessary. This tree will not have a staight trunk if you keep it, because the small branch will be forced to form a new leader.
I have a lot of aggressive-growing, mature vines along that area - I'm thinking maybe they were depriving the trees of water and/or nutrients.
This is probably not the main reason for your trees' poor health. The vines should be cut back if they attempt to climb the trees. It sounds like the trees were damaged in the winter, from the above ground parts of the trees drying out when the roots were frozen. There have been many similar cases this spring with newly planted trees, even in my area (zone 6b), because of the fairly harsh winter.
I have the same issue with 2 Douglas Fir trees I planted about 3 years ago. Other than the fact that they've barely grown in the 3 years, this spring 2 of the 3 have lost all their needles.
It sounds like the trees never came out of transplant shock. This can occur if the hole was too small, and the roots weren't spread out properly during planting. Also, compacted clay soil can do this, if the sides of the planting hole were not loosened. If the soil is shallow and nutrient-deprived, the trees will find it hard to put out root growth, and therefore top growth. Evergreens also need to be watered well before winter in dry areas, because they will let off moisture all winter, even when the roots cannot supply more.
That is what happened to the "two of the three" which "have lost all their needles". Those two are dead and can be removed. If the remaining tree is growing, it should be mulched well and fertilized, and given a good bit of water if/when dry. If the tree is still green, but hasn't budded out, things look grim You can try transplanting if you have had any of the planting problems mentioned above. I'm assuming from: "they were completely submersed in snow for awhile" that they were not balled and burlapped.
Dig the hole first, making sure to dig a good bit wider than the roots of the tree will be, spread out evenly. Make sure you have enough good fill soil nearby. You want to use a soft, free-draining, acidic soil with a high organic matter content. Dig the tree out carefully, to get out all the roots with as little damage as possible. Plant it as soon as you can, making sure the top of the root flare is even with the surface (even if it wasn't before). Do not let it dry out.